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Asymptotic Variance of Newton-Cotes Quadratures based on Randomized Sampling Points

By Mads Stehr and Markus Kiderlen
CSGB Research Reports
No. 02, February 2019

In this paper we consider the problem of numerical integration when sampling nodes are random, and we suggest to use Newton-Cotes quadrature rules to exploit smoothness properties of the integrand. In previous papers it was shown that a Riemann sum approach can cause a severe variance inflation when the sampling points are not equidistant. However, under some integrability conditions on the typical point-distance, we show that Newton-Cotes quadratures based on a stationary point process in $\mathbb{R}$ yield unbiased estimators for the integral and that the aforementioned variance inflation can be avoided if a Newton-Cotes quadrature of sufficiently high order is applied. In a stereological application, this corresponds to the estimation of volume of a compact object from area measurements on parallel sections.

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